Question

Is `f(x)=-9x^5+15x^4-21x^2-8x+2` concave or convex at `x=3`?

Answer 1

concave at x = 3

Explanation:

To test for concavity/convexity , require to evaluate f''(x).

• If f''(x) > 0 then f(x) is convex

• If f''(x) < 0 then f(x) is concave

require to find f''(x) and then value of f''(3)

`f'(x) = - 45x^4 + 60x^3 - 42x - 8`

`f''(x) = -180x^3 + 180x^2 - 42`

`f''(3) = -180(3)^3 + 180(3)^2 -42`

= -4860 + 1620 - 42 = - 3282 < 0

since f''(3) < 0 then f(x) is concave at x = 3